1 Chapter XII. Mathematics - general part and standards Article 61. Mathematics - General Part Introduction In modern times, mathematics is an integral part of life. It is used in all areas of human activity: in science and technology, in medicine, in economics, in the protection and restoration of the environment, in the adoption of social decisions. It is also worth mentioning the role of mathematics in the development of humanity and the establishment of modern civilization. Developing digital technologies, better understanding of the space-time structure, discovering many regularities of the nature clearly illustrates the scientific and cultural value of mathematics. The important thing is that, mathematics promotes the development of human mental abilities. It provides an effective, laconic and unambiguous communication. Mathematics is used to illustrate difficult situations, explanation of events and foreshadow their results. Abstract systems and theoretical models created in mathematics are used to study regularity, analyze the situation and solve problems. In the solution of the problem it is necessary to access its entity, select adequate mathematical apparatus, and in the absence of such a task - to develop it, to create a studied process or object- oriented model, to make the conclusions conveyed through the received model and then interpret. Practical and scientific problems, in turn, provide mathematics with important and interesting tasks. Consequently, the importance of teaching mathematical methods should be taken into consideration during the study, in the management of social-economic or technical processes, solving household and scientific problems, and forming a logically organized system and imparting this knowledge to others. Additionally, in math learning, focusing on the practical as well as the scientific problem is the ability to motivate pupils and arise the interest in mathematics. Goals and objectives of math teaching The basic purposes of teaching mathematics in the institution of general education are the following: Establishing a pupil as an active citizen, who will be able to investigate and analyze the issues related to the real situation, be creative and organized; Creating profitable background for the development of mathematical science in the country. Mathematical knowledge means possessing mathematical concepts and procedures, the ability to use them in solving real problems; As well as the means of communication that are needed to obtain and transmit information using mathematical language and means. Due to educational aims, the curriculum envisages solving of specific tasks. Such tasks are as follows:
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Chapter XII. Mathematics - general part and standards
Article 61. Mathematics - General Part
Introduction
In modern times, mathematics is an integral part of life. It is used in all areas of human activity: in
science and technology, in medicine, in economics, in the protection and restoration of the
environment, in the adoption of social decisions. It is also worth mentioning the role of
mathematics in the development of humanity and the establishment of modern civilization.
Developing digital technologies, better understanding of the space-time structure, discovering
many regularities of the nature clearly illustrates the scientific and cultural value of mathematics.
The important thing is that, mathematics promotes the development of human mental abilities. It
provides an effective, laconic and unambiguous communication. Mathematics is used to illustrate
difficult situations, explanation of events and foreshadow their results. Abstract systems and
theoretical models created in mathematics are used to study regularity, analyze the situation and
solve problems.
In the solution of the problem it is necessary to access its entity, select adequate mathematical
apparatus, and in the absence of such a task - to develop it, to create a studied process or object-
oriented model, to make the conclusions conveyed through the received model and then interpret.
Practical and scientific problems, in turn, provide mathematics with important and interesting
tasks. Consequently, the importance of teaching mathematical methods should be taken into
consideration during the study, in the management of social-economic or technical processes,
solving household and scientific problems, and forming a logically organized system and imparting
this knowledge to others. Additionally, in math learning, focusing on the practical as well as the
scientific problem is the ability to motivate pupils and arise the interest in mathematics.
Goals and objectives of math teaching
The basic purposes of teaching mathematics in the institution of general education are the
following:
Establishing a pupil as an active citizen, who will be able to investigate and analyze the
issues related to the real situation, be creative and organized;
Creating profitable background for the development of mathematical science in the
country.
Mathematical knowledge means possessing mathematical concepts and procedures, the ability to
use them in solving real problems; As well as the means of communication that are needed to
obtain and transmit information using mathematical language and means.
Due to educational aims, the curriculum envisages solving of specific tasks. Such tasks are as
follows:
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Developing skills of thinking, discussing, approving the opinions, developing the ability of events
and facts analysis for pupils:
Conjecturing the assumption and its research in private cases;
Selecting and organizing the initial data;
Selecting the way of approval, confirming;
Considering the usefulness of the chosen strategy and utilization of its borders;
Development of the line of arguing, searching alternative ways, justifying the correctness
and efficiency of the received decision;
Mastering mathematics as a universal language of description of the universe and science:
Evaluation of the place and importance of mathematics in different disciplines, in business,
arts, and human activities;
To observe the mathematical objects and processes in the usual environment (daily life)
and use their properties when modeling, solving practical (present) tasks;
Mastering and using the means and methods of presenting the information; making an
interpretation of the information presented in various ways, discuss about them and
connect them with each other;
Understanding and analyzing others' perceptions; presenting the essence of the matter in
the transmission of information.
Preparing for the next level of study or professional activities; imparting the necessary
knowledge to solve the life problems and develope the skills to use the very knowledge:
Mastering the ways and methods of arranging and planning the work;
Perciving the task content, understanding-marking off and analyzing the tasks’ data and
index quintities;
Determining the problem and formulating it;
Solving the complex problem gradually, dividing it into simple tasks and in stages;
Selecting and using the necessary strategies and resources to solve the problem;
Critical assessment of the obtained result taking into account the context;
Selecting and using the adequate supporting techniques and technologies in solving the
problem.
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Arranging the Mathematics Teaching
Mathematics is taught in each class of each level of general education as a compulsory subject.
Evaluation in mathematics
In order to follow the standard requirements, it is recommended to use a wide variety of
mathematical tasks to evaluate pupils. The types of tasks can be as follows:
1. To read the text of the mathematical content (including the text containing the diagrams and
tables), make a conclusion by the received data analysis (based on calculations or logical reasoning)
and substantiate it;
2. The task in which a pupil is required to prove or deny the given fact of the given (e.g. theorem
approval);
3. To calculate the value of numerical symbol, solve the equation, simplify the lettering image;
select the correct answers between several possible responses of the task, establish compliance,
sort the data in a sequence;
4. Geometric task in which the pupil is required to determine the characteristics of the divided part
of the figure, define the measures, draw divided parts of a figure;
5. To draw a diagram or graphic, make a spatial slips or model, and so forth.
The components of mathematical task evaluation are as follows:
1. To know and use mathematical concepts and regulations;
2. To establish links and relationships between mathematical objects;
3. To identify mathematical objects and mathematical language;
4. To argue - approve;
5. To present the essence of the matter in the transmission of information;
6. To establish a task;
7. To divide the complex task into simple ones in stages;
8. Mathematical modeling;
9. Finding and realizing the way of solving the problem;
10. Choosing the optimal method of calculation; Accuracy of calculating implementation;
11. To use auxiliary technical means and informational technologies.
The pupil’s living skills are included in the evaluative focus:
1. Creativity;
2. Cooperation (with partner, group members);
3. Intelligible use of strategies for promoting educational activities;
4. Quality of participation in study activities.
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Note: In assessing students in primary classes, special attention will be paid to the following
skills:
1. Writing down and naming the numbers;
2. Performing arithmetic operations and verbal description;
3. Identify and describe the geometric figures;
4. Designing the figures;
5. Knowing and using the measuring a distance and defining facilities / ways for it;
6. Verbal and schematic description of the direction, movement and route;
7. Terms: "all", "every", "each", "some", "one", "none", "only" are used when determining the
connection between the numbers or the numbers’ unity;
8. Classifying, grouping and sorting the data according to the pointed criteria;
9. Use measuring units and know (distance, time, money units) the relationship between them.
The component of summary tasks
The component of the summary tasks is related to the study-teaching results. This component
should evaluate the results achieved through the study and processing of one study section (topic,
chapter, paragraph, issue). Upon completion of a particular study unit, the pupil should be able to
demonstrate the knowledge and skills defined by the program of mathematics. Consequently, the
summary tasks should evaluate the results defined by the Mathematical Subjects Program.
Types of summary tasks:
To follow the standard requirements, it is recommended to use the various forms of summary
tasks. The types of mathematical summary tasks can be as follows:
1. Choosing open or closed type of work (the correct answer between several possible responses,
establishment of conformity and correct order) related to the textual task;
2. To read the text, convey the obtained conclusion by the result of the data analyses (including
text, containing diagrams and tables) and approve it (based on calculations or logical reasoning);
3. To solve equation, simplify lettering, calculate the index quantity of numerical image;
4. Geometric task in which the pupil is required to determine the characteristics of the figure,
define the measures, construct the figure;
5. The task in which the pupil is required to approve or deny the fact based on the data defined in
advance (e.g. theorem approval).
The requirements, which should be satisfied by the summary evaluations:
1. Each type of task should be accompanied by a general rubric of its evaluation;
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2. General rubric should be approved in accordance with the specific work condition and learned
material;
It should be indicated those results of the standard, for the evaluations of which the summary
work is defined.
Article 62. Mathematical Standard
Mathematics on the primary level
Introduction
The Mathematical Curriculum has four directions: numbers and operations; geometry and
perception of space; data analysis, statistics and probability; Regularities and algebra.
These trends are in close relationship and include knowledge and skills that the pupil must acquire
in general education. The division of the directions does not mean a similar division of the course,
it only shows the spectrum of the study material and gives the possibility to determine to what
direction the attention should be paid on this or that level of teaching.
1. Direction: Numbers and operations
Numbers, their utilization and the means of numbers’ representation;
Operation numbers and numerical correlations;
Assessment and approximation of number;
Quantities, size units and other utilization of numbers.
The main objectives of this direction are to develop "number sense", master the calculative
principles, study arithmetic operations and their properties, master methods of calculation and
exploitation of the results of the evaluation of the number; study mutual comparison and
utilization of the arithmetic operations during the process of their fulfillment and estimation of the
practical tasks; learn the number system.
On the primary level, it should be developed the arithmetic operations and their adequate
utilization; understanding the properties of arithmetic operations and their connections;
developing the ability to evaluate the effect of the arithmetic operations and the importance of
numerical image.
In addition, the pupil should be able to develop a comprehensive understanding of the decimal
positional system and the ability to use it when performing multi-digit numbers; the perception of
the whole parts (half, third, fourth), the ability to demonstrate, construct and compare them.
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2. Direction: Geometry and perception of space
Geometric objects: their properties, interaction and constructing;
Size and means of measurement;
Transformations and symmetry of figures;
Coordinates and their using in geometry.
The main purpose of this direction is to study geometric objects and their qualities, measurements,
geometrical transformations and geometric objects, algebraic methods, present applied aspects of
the geometry.
On the primary level the main objective of the direction is to develop the ability of depicting the
mutual deployment of geometric objects and demonstrating them; the ability to identify the
components of geometric objects and describe their interaction; recognize the figure and create
the model according to the attributes of the figure-group, verbal descriptions, develop the ability
of making evaluation and in the standard units measurement of the length of a distance between
the two points, the skill of description and schematic imagery of the route.
3. Direction: Analysis, probability and statistic of data
Data sources and means of obtaining data sources;
Data regulation methods and means of data representation;
Summarizing numerical characteristics of data;
Probable models;
Selective method and numerical characteristics of the selection.
The aim of the introduction of statistical concepts and apparatus in the general educational
establishment is to regulate the pupil's intuitive representations about the data, develop the skills
of establishing the intuition of the pupils in the structural way and the ability of using the
probabilistic statistical methods.
The aim of the direction on the primary level is to let the pupils to get acquainted with the
elements of descriptive statistics - the means of collecting, regulating, presenting, and interpreting
qualitative and discreet quantitative data.
4. Direction: Regulations and Algebra
Sets, reflections, functions and their use;
Discrete math elements and their use;
Algorithms and their use;
Algebraic operations and their properties.
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The main objective of this direction is to make a pupil develop the ability of identifying and
describing the regulations, algebraic relationships and functional attitudes, as well as the ability of
modeling and solving problems by them.
The aim of the direction on the primary level is to develop the ability to recognize dependence
between the ordinary regularities and quantities, study the use of arithmetic operations and
lettering indications.
The Standard of I-IV class
a) The results to be achieved at the end of the IV class
Definition of indices
On the primary level each crossed off outcome of the standard is preceded by the index, which
indicates the subject, the level of teaching and the number of the standard results; for example,
math.pr. (I).1:
"math" - indicates the subject of "mathematics";
“pr. " - indicates the primary levels;
"(I)" - indicates that the standard includes I-IV classes;
"1" - indicates the resulting number.
Mathematics Standard (I-IV Classes)
Results Index Standard results
1. Direction - Numbers and operations
Sub directions
numbers, using and presenting number means
math.pr.(I) 1 A pupil should master and use the concept of the number and present the means of number; He must be able to compare and classify numbers.
operations on numbers and numerical ratios
math.pr.(I).2 A pupil should be able to make basic arithmetic operations on numbers.
Number evaluation and rounding off
math.pr.(I).3 A pupil should be able to evaluate the results of the actual effect on the natural numbers and round the numbers off.
Values, size units other using of numbers
math.pr.(I).4 A pupil should be able to use different units of the size and connect with each other.
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2. Direction - Geometry and perception of space
Sub directions A pupil should be able to:
Geometric objects: their properties, interconnecting and constructing them
math.pr.(I).5 make identification, classification, depiction of figures.
Size and measurement means
math.pr.(I).6 find the distance between the dimensions of the figures, the subjects and the objects.
Coordinates and their use in geometry
math.pr.(I).7 orientate on the plane.
3. Direction - Regularities and Algebra
Sub directions A pupil should be able to:
Sets,depicting, functions and their use
math.pr.(I).8 extend, depict and analyze the given correspondence between the subjects or the subjects and their attributes.
Algebraic operations and their properties
math.pr.(I).9 draw the equality containing the number image and its use for solving the problem.
4. Direction - Data Analysis, Probability and Statistics
Sub directions
Data sources and means of data obtaining
math.pr.(I).10 A pupil should have basic ideas about the means of obtaining information.
Data regulating methods and means of data representation
math.pr.(I).11 A pupil should master the ways/means of the information arrangement and display
Data summarizing numerical characteristics
math.pr.(I).12 A pupil should be able to interpret qualitative and quantitative data and elementary analysis.
5. Direction – Discusing-Substantiating, Problem Solving
Sub directions
Discusing-Substantiating Solving problem
math.pr.(I).13 A pupil should be able to solve simple tasks concerning everyday life or when it is caused by the field of natural science.
B) content
1. The concept of number. Use of numbers (e.g., notch, row);
2. Natural numbers within a million. Decimal positioning system. Simplistic introduction on other
numerical systems (numerical alphabets and letter numbers, roman system);
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3. Arithmetic operations on natural numbers. Divide with the remainder. Comparing numbers and
evaluating the results of arithmetic operations. Features of arithmetic operation: Commutation of
addition and multiplication (transferability), associativity (pairing off numbers) and distributivity
towards multiplication (schaduling);
4. Textual tasks that are explained by numerical depiction of addition, subtraction, multiplication
and division;
5.To measure the units, connections between units and using size units;
6. Geometric figures (flat, spatial). Figure elements and depiction;
7. Classification of figures with different marks (e.g. flat and spatial);
8. Figure dimensions: measuring tools and length measuring units. Perimeter of polygon;
9. Spatial shapes models, divided parts of a figure.
10. General representations of the qualitative and quantitative data (task):gathering means: to
measure, observe, survey; Data picking from the simplest sources (e.g. from the reference book);
11. Organizing qualitative and quantitative data: data grouping; sorting of quantitative data on
growth and convergence; Quality data ordering (e.g. with a lexicographic method);
12. The means of data presentation: table, pictogram, diagram;
13. Summarizing numerical data for qualitative and quantitative data: the total number of data, the
greatest and the smallest values.
The V-VI classes standard
a) The results to be achieved at the end of the VI class
Definition of indices
Each crossed off outcome of the standard on the primary levels is preceded by the index, which
indicates the subject, the level of teaching and the number of the standard results; for example,
math.pr. (II).1.:
"math" - indicates the subject of "mathematics";
“pr." - indicates the primary levels;
"(II)" - indicates that the standard includes V-VI classes;
"1" - indicates the resulting number.
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Mathematics Standard (V-VI Classes)
Results Index
Standard results
1. Direction: Numbers and activities
Sub directions A pupil should be able to:
Numbers, their use and
the means to represent
them.
math.pr.(II).1. depict, classify and use of natural and nonnegative rational
numbers.
Operations on the
numbers and numerical
correlations.
math.pr.(II).2. perform arithmetic operations on natural and nonnegative
rational numbers.
Evaluation of the quantity
and approximation.
math.pr.(II).3. evaluate approximately the result of operations on natural
numbers and round the numbers off.
Values, size units and
other use of numbers.
math.pr.(II).4. Connect and use different units of the size.
2. Direction: Geometry and perception of space
Sub directions A pupil should be able to:
Geometric objects: their
properties, their
interconnection and
construction.
math.pr.(II).5. identify, describe, and depict flat and spatial geometric
figures and establish relations between the figures and the
elements of the figure.
Size and measurement
means.
math.pr.(II).6. calculate a flat figure space and use it in the tasks according